Chaos: Not Quite (but Almost) Randomness – Part 1 of 2

Is our solar system stable, or will the orbits of the planets at some point collapse into the Sun? Closer to home: will it rain tomorrow?

Both these questions turn out to be surprisingly tricky to answer for the same underlying reason: the mathematical models we use to understand these systems are chaotic.

Chaotic systems are not random, but exhibit behavior so unpredictable as to appear random, stemming from great sensitivity to small changes in initial conditions. Meteorologist Philip Merilees memorably characterized weather systems as chaotic when he wrote of “the flap of a butterfly’s wings in Brazil [setting] off a tornado in Texas”: the butterfly causes a small change in the local atmosphere, and the larger-scale behavior of the weather changes drastically as a result.

If we are trying to understand some system—our solar system or the weather—using a mathematical model, chaos in the system makes accurate predictions difficult because even tiny errors in the input parameters to the model can balloon into huge errors in the predictions output by the model. This phenomenon can be seen in Figure 1: starting from two points which are very close, but not the same, the system eventually takes very different paths.

Figure 1: Illustration of two diverging trajectories (paths) originating from nearby points. The arrows on the left show where each trajectory starts; the other pair of arrows show where they end up after the same amount of time. These trajectories are from a Lorenz system—a model that Edward Lorenz used to study weather systems. (Here is another illustration of the same phenomenon.) Image from Wikimedia Commons.

Lorenz saw this vividly one day in 1961, when he was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full-precision value of 0.506127. The result was a completely different weather scenario.

The Three-Body Problem

The weather on Earth may be more directly relevant to our daily lives, but the first chaotic system to be studied mathematically was the solar system—or, rather, a simplified version involving just three celestial bodies.

The motion of the planets (and their moons, and the Sun) is governed—to a first approximation –by Newton’s laws of motion and universal gravitation. These laws, described in the late 17th century, have succinct, elegant mathematical formulations. Can we deduce an equally succinct description of the motion of the planets—or of any such collection of several masses—using these laws? Such a description would allow us to definitively establish the stability of the solar system.

This is commonly called the “N-body problem“, N being the number of masses. Surprisingly, it has proven fiendishly difficult, even for just N=3.

I Got Infinite Problems…

In principle, the solution to the N-body problem should be straightforward. All we have to do is solve an explicit system of equations where the variables are the positions and momenta of the masses involved. In practice, however, the computations get immensely complicated once N is bigger than 2: the motion of each mass is affected by the gravity of each other mass, whose motion is in turn affected by the gravity of each other mass, and so on, and so on … ad infinitum.

How are we to figure anything out if all the computations are infinite? One thing we could do is just stop somewhere along the line and say, “that’s good enough.” We didn’t add up all the contributions, but maybe the remaining ones are small enough that they don’t affect the outcome very much.

As an analogy: 1/2 + 1/4 + 1/8 + … is a sum of infinitely many terms, but, as we add up more and more of them, the sums we get don’t shoot off to infinity. Instead, they get closer and closer to 1 (Figure 2).

This sum is an example of a convergent infinite series, and convergence is one of the main tools we have for getting a practical handle on otherwise “infinite” solution processes. The theory of infinite series was worked out and put on firm mathematical foundations in the 19th century. The hope then arose that the N-body problem might have a solution in terms of convergent series.

In 1887, in honor of his 60th birthday, Oscar II, King of Sweden, established a prize for anyone who could find such a solution to the N-body problem. Twelve papers were submitted for the competition. Only five of them addressed the n-body problem as stated, and none of them could obtain a solution in uniformly convergent series. Under the circumstances, the jury awarded the prize to French polymath Henri Poincaré, whose submission claimed to solve the 3-body problem using a series of stable approximations to the orbits. These were trajectories approximating the orbits, which do not change significantly under small changes to the initial conditions—unlike those illustrated in Figure 1, for example.

After he had received the prize and his submission had been accepted for publication, Poincaré realized he had made a mistake and that his simplifications did not indicate a stable orbit after all. In fact, he realized that even a very small change in his initial conditions would lead to vastly different orbits—the hallmark of a chaotic system.

Since we do not know, even today, the state of the solar system with infinite precision, we cannot, in fact, know with absolute certainty if our solar system is stable!

This is disheartening; nevertheless, in the second part of this post we will find out how to deal with chaos and get a handle on chaotic systems such as our solar system.

About the author

Feng Zhu just finished his fourth year in the Math PhD program at the University of Michigan. He studies geometry and topology, or more specifically the weird and wonderful things that groups of symmetries of negatively-curved spaces can do. He was born in Shanghai, China, grew up mostly in Singapore, went to college in Princeton, New Jersey, and finally came to Ann Arbor for graduate school. (In short, he realized that the tropics aren’t all that cool and started migrating back towards the cold.) When not doing math, he enjoys running, traveling, reading, attempting to learn languages, and playing the keyboard.